Methods for computation-free wideband spectral correlation and analysis

ABSTRACT

An apparatus for generating a set of spectral correlation coefficients of an input signal includes: a master laser configured to generate an optical frequency comb signal; a first optical modulator configured to modulate the optical frequency comb signal with an input signal to generate a plurality of spectral copies of the input signal; a dispersive element configured to delay the plurality of spectral copies of the input signal by a wavelength-dependent time delay; a second optical modulator configured to modulate the delayed plurality of spectral copies with a conjugate of the input signal; and an optical comb filter configured to integrate the conjugate modulated plurality of spectral copies of the input signal to generate a set of cyclic autocorrelation coefficients.

CROSS-REFERENCES TO RELATED APPLICATIONS

This application claims the benefit of U.S. Provisional Application No.62/666,363, filed May 3, 2018, the contents of which are herebyincorporated by reference in their entirety for all purposes.

BACKGROUND OF THE INVENTION

Techniques for wideband spectral correlation function and/orcyclic-autocorrelation function estimation have been developed. Despitethe progress made in digital techniques for spectral correlationcalculation that is done in digital electrical domain, the intensivecomputation required for wideband signal analysis is beyond state of theart electronics capability. There is a need in the art for improvedmethods and systems related to spectral correlation calculation.

SUMMARY OF THE INVENTION

In accordance with various aspects of the present disclosure,apparatuses and methods for computation-free or reduced computationwideband spectral correlation and analysis are provided.

According to various aspects, there is provided an apparatus forgenerating a set of cyclic autocorrelation coefficients of an inputsignal. In some aspects, the apparatus may include: a master laserconfigured to generate an optical frequency comb; a first opticalmodulator configured to modulate the optical frequency comb with aninput signal to generate a plurality of spectral copies of the inputsignal; a dispersive element configured to delay the plurality ofspectral copies of the input signal by a wavelength-dependent timedelay; a second optical modulator configured to modulate the delayedplurality of spectral copies with a conjugate of the input signal; andan optical comb filter configured to integrate the conjugate modulatedplurality of spectral copies of the input signal to generate the set ofcyclic autocorrelation coefficients.

The apparatus may further include: a local oscillator configured togenerate a swept frequency; a third optical modulator configured tomodulate an optical signal generated by the master laser with the sweptfrequency from the local oscillator to generate swept optical samplingsignals; a wavelength demultiplexer configured to combine the sweptoptical sampling signals and the integrated conjugate modulated delayedplurality of spectral copies of the input signal such that each of theswept optical sampling signals samples a corresponding integratedconjugate modulated delayed spectral copy of the input signal; and aplurality of detectors configured to detect the set of cyclicautocorrelation coefficients for all time delays of an associated tonefrom the integrated conjugate modulated delayed spectral copies of theinput signal, and output the detected cyclic autocorrelationcoefficients.

According to various aspects, there is provided a method for generatinga set of cyclic autocorrelation coefficients of an input signal. In someaspects, the method may include: generating an optical frequency combsignal from a master laser optical signal; generating a plurality ofspectral copies of the input signal by modulating the optical frequencycomb signal with the input signal; delaying the plurality of spectralcopies by a wavelength-dependent time delay; modulating the delayedplurality of spectral copies with a conjugate of the input signal; andgenerating the set of cyclic autocorrelation coefficients by integratingthe conjugate modulated delayed plurality of spectral copies of theinput signal.

The method may further include: generating a swept frequency; modulatingan optical signal from the master laser with the swept frequency togenerate swept optical sampling signals; combining the swept opticalsampling signals and the integrated conjugate modulated delayedplurality of spectral copies of the input signal such that each of theswept optical sampling signals samples a corresponding integratedconjugate modulated delayed spectral copy of the input signal; detectingthe set of cyclic autocorrelation coefficients for all time delays of anassociated tone from the integrated conjugate modulated delayed spectralcopies of the input signal; and outputting the detected cyclicautocorrelation coefficients.

According to various aspects, there is provided an apparatus forgenerating coefficients for a Wigner function in an optical domain. Insome aspects, the apparatus may include: a master laser configured togenerate an optical frequency comb signal; a first optical modulatorconfigured to modulate the optical frequency comb signal with a firstsignal to generate a plurality of spectral copies of the first signal; adispersive element configured to delay the plurality of spectral copiesof the first signal by a wavelength-dependent time delay; a secondoptical modulator configured to modulate the delayed plurality ofspectral copies with a second signal; and an optical comb filterconfigured to integrate the modulated delayed plurality of spectralcopies of the first signal to generate a set of Wigner functioncoefficients.

The apparatus may further include: a local oscillator configured togenerate a swept frequency; a third optical modulator configured tomodulate an optical signal generated by the master laser with the sweptfrequency from the local oscillator to generate swept optical samplingsignals; a wavelength demultiplexer configured to combine the sweptoptical sampling signals and the integrated modulated delayed pluralityof spectral copies of the first signal such that each of the sweptoptical sampling signals samples a corresponding integrated modulateddelayed spectral copy of the first signal; and a plurality of detectorsconfigured to detect the set of Wigner function coefficients for alltime delays of an associated tone from the integrated modulated delayedspectral copies of the first signal, and output the detected set ofWigner function coefficients.

According to various aspects, there is provided a method for generatinga set of coefficients for a Wigner function in an optical domain. Insome aspects, the method may include: generating an optical frequencycomb signal from a master laser optical signal; generating a pluralityof spectral copies of a first signal by modulating the optical frequencycomb signal with the first signal; delaying the plurality of spectralcopies by a wavelength-dependent time delay; modulating the delayedplurality of spectral copies with a second signal; and generating theset of coefficients for the Wigner function by integrating the delayedmodulated plurality of spectral copies of the first signal.

In an embodiment, the optical frequency comb signal and the delayedplurality of spectral copies are amplitude modulated. In anotherembodiment, the second signal is a conjugate of the first signal and theWigner function is a cyclic autocorrelation function. Additionally, thesecond signal can be a conjugate of a signal different from the firstsignal received from a same source as the first signal and the Wignerfunction can be a cross-ambiguity function.

The method may further include: generating a swept frequency; modulatingan optical signal generated by the master laser with the swept frequencyto generate swept optical sampling signals; combining the swept opticalsampling signals and the integrated modulated delayed plurality ofspectral copies of the first signal such that each of the swept opticalsampling signals samples a corresponding integrated modulated delayedspectral copy of the first signal; detecting the set of coefficients forthe Wigner function for all time delays of an associated tone from theintegrated modulated delayed spectral copies of the first signal; andoutputting the detected set of Wigner function coefficients. The opticalsignal generated by the master laser can be single side band modulated.Also, detecting the set of coefficients for the Wigner function caninclude coherent detection.

Numerous benefits are achieved by way of the various embodiments overconventional techniques. For example, the various embodiments providemethods and systems that can be used to circumvent basic limits imposedby the analog-to-digital converter (ADC) and repeated Fast-FourierTransform (FFT) used in a conventional cyclostationary (CS) receiver. Insome embodiments, radio frequency (RF) signal analysis may be performedin the optical domain, thereby eliminating the need for a high-speed ADCand requiring less processing power for computing FFTs. These and otherembodiments along with many of its advantages and features are describedin more detail in conjunction with the text below and attached figures.

BRIEF DESCRIPTION OF THE DRAWINGS

Aspects of the present disclosure now will be described more fullyhereinafter with reference to the accompanying drawings, which areintended to be read in conjunction with both this summary, the detaileddescription and any preferred and/or particular embodiments specificallydiscussed or otherwise disclosed. The various aspects may, however, beembodied in many different forms and should not be construed as limitedto the embodiments as set forth herein; rather, these embodiments areprovided by way of illustration only and so that this disclosure will bethorough, complete and will fully convey the full scope to those skilledin the art.

FIG. 1A is an illustration of a spectral correlation function S_(x) ^(α)for a cyclostationary minimum shift keying (MSK) modulated signal havinga signal-to-noise ratio (SNR) of 0 dB;

FIG. 1B is an illustration of the cyclic autocorrelation function R_(x)^(α) corresponding to the cyclostationary signal of FIG. 1A;

FIG. 1C is an illustration of a spectral correlation function S_(x) ^(α)for a cyclostationary minimum shift keying (MSK) modulated signal havinga signal-to-noise ratio (SNR) of 10 dB;

FIG. 1D is an illustration of the cyclic autocorrelation function R_(x)^(α) corresponding to the cyclostationary signal of FIG. 1C;

FIG. 2 is a block diagram of a photonic assisted cyclic autocorrelationprocessor in accordance with various aspects of the present disclosure;

FIG. 3 is a block diagram of a cyclic autocorrelation coefficientsreadout circuit in accordance with various aspects of the presentdisclosure;

FIG. 4 is a block diagram of a signal conjugate generator in accordancewith various aspects of the present disclosure;

FIG. 5 is a flowchart of a method for generating cyclic autocorrelationcoefficients in accordance with various aspects of the presentdisclosure;

FIG. 6 is a flowchart of a method for reading out cyclic autocorrelationcoefficients in accordance with various aspects of the presentdisclosure;

FIG. 7 is a block diagram of a two pumps parametric mixer embodiment ofa photonic assisted cyclic autocorrelation processor in accordance withvarious aspects of the present disclosure;

FIG. 8 is a diagram illustrating an example application of thecross-ambiguity function in accordance with various aspects of thepresent disclosure;

FIG. 9 is a block diagram of a photonic assisted cross-ambiguitycorrelation processor configured for determining a cross-ambiguityfunction in accordance with various aspects of the present disclosure;

FIG. 10 is a flowchart of a method for generating cross-ambiguitycorrelation coefficients in accordance with various aspects of thepresent disclosure;

FIG. 11 is a flowchart of a method for reading out cross-ambiguitycorrelation coefficients in accordance with various aspects of thepresent disclosure;

FIG. 12 is a block diagram of a two pumps parametric mixer embodiment1200 of a photonic assisted cyclic autocorrelation processor configuredfor determining a cross ambiguity function (CAF) in accordance withvarious aspects of the present disclosure; and

FIG. 13 is a flowchart of a method for generating and reading out Wignerfunction coefficients in accordance with various aspects of the presentdisclosure.

DETAILED DESCRIPTION OF SPECIFIC EMBODIMENTS

The present invention relates generally to wideband spectral correlationanalysis, and in particular to computation-free or reduced computationwideband spectral correlation analysis. The architecture of the presentdisclosure was developed to circumvent basic limits imposed by the ADCand DFT in conventional CS receivers. Cyclostationary and high-order(cumulant) analysis is based on the fact that signals and noise havedifferent correlation (moment) properties; however, to realize this, onedoes not necessarily need time-to-frequency mapping.

Spectral correlation has been used in signal detection, for example intelecommunications systems. The spectral correlation function of acyclostationary process describes the cross-spectral density, orcoherence, of all pairs of frequency-shifted versions of a time-series.Calculation of the spectral correlation causes the stochastic portion(i.e., noise) of the cyclostationary process to vanish while thedeterministic portion having cyclic features emerges.

Cyclostationary and high-order (cumulant) analysis is based on a factthat signals and noise have different correlation (i.e., moment)properties. A modulation-bearing signal x(t) possesses a unique cyclicautocorrelation defined as the Fourier transform of a signal and itsdelayed copy as shown in equation (1):

$\begin{matrix}{{{R_{x}^{\alpha}(\tau)} = {\int_{- \infty}^{\infty}{{x\left( {t - \frac{\tau}{2}} \right)}{x^{*}\left( {t + \frac{\tau}{2}} \right)}e^{{- {j2{\pi\alpha}}}\; t}{dt}}}}\ } & (1)\end{matrix}$

One of ordinary skill in the art will recognize equation (1) as a Wignerfunction that approximates how the spectral density changes in time. Thespectral correlation function is the Fourier counterpart of the cyclicautocorrelation R_(x) ^(α)(τ) as shown in equation (2):

S _(x) ^(α)(f)=∫_(−∞) ^(∞) R _(x) ^(α)(τ)e ^(−j2πfτ) dt  (2)

Rather than generating the cyclic autocorrelation function R_(x)^(α)(τ), conventional cyclostationary (CS) receivers first map adiscretized signal via discrete Fourier transform (DFT) as shown inequation (3):

$\begin{matrix}{{X\left( {n,k} \right)} = {\sum_{r = {{{- N}/2} + 1}}^{N/2}{{x\left( {n - r} \right)}e^{\frac{{- {j2\pi}}\; {k{({n - r})}}}{N}}}}} & (3)\end{matrix}$

and then produce the spectral correlation function S_(x) ^(α) shown inequation (4) by populating a spectral product table:

$\begin{matrix}{{S_{x}^{\alpha}\left( {n,k} \right)} = {\frac{1}{M}{\sum_{n = 0}^{M - 1}{\frac{1}{N}{X\left( {n,{k + \frac{\alpha}{2}}} \right)}{X^{*}\left( {n,{k - \frac{\alpha}{2}}} \right)}}}}} & (4)\end{matrix}$

The frequency of a signal that can be processed in this manner iscomputationally intensive and is limited by the bandwidth and resolutionof the analog-to-digital converter (ADC) used to create the discretesignal as well as the processing speed for performing the repeated DFTand multiplications.

FIG. 1A is an illustration of an example of a spectral correlationfunction S_(x) ^(α) for a cyclostationary minimum shift keying (MSK)modulated signal having a signal-to-noise ratio (SNR) of 10 dB. Thesignal has a 2 Gbps data rate and is upconverted at 8 GHz. FIG. 1B is anillustration of the cyclic autocorrelation function R_(x) ^(α)corresponding to the cyclostationary signal of FIG. 1A. FIG. 1C is anillustration of an example of a spectral correlation function S_(x) ^(α)for a cyclostationary minimum shift keying (MSK) modulated signal havinga signal-to-noise ratio (SNR) of 0 dB. The signal has a 2 Gbps data rateand is upconverted at 8 GHz. FIG. 1D is an illustration of the cyclicautocorrelation function R_(x) ^(α) corresponding to the cyclostationarysignal of FIG. 1C. In FIGS. 1A and 1C the frequency (f) axis is scaledfrom −10 GHz to 10 GHz with f=0 at the center. Similarly the cyclicfrequency axis (a) is scaled from 10 GHz to −10 GHz with α=0 at thecenter. In FIGS. 1B and 1D the time (i.e., delay) axis (τ) is scaledfrom τ=−1/f to τ=1/f with τ=0 at the center, and the cyclic frequencyaxis (a) is scaled from 10 GHz to −10 GHz with α=0 at the center.

Both the Spectral Correlation functions S_(x) ^(α) in FIGS. 1A and 1Cand the cyclic autocorrelation functions R_(x) ^(α) in FIGS. 1B and 1Dreveal unique modulation signatures by discriminating noise. FIGS. 1A-1Dshow the modulated signal buried in noise: S_(x) ^(α) localizes noisealong the ridge at α=0, while R_(x) ^(α) confines the noisesubstantially to the center of the map at τ=0 and α=0. As can be seen inFIGS. 1A-1D, by comparing the spectral correlation function S_(x) ^(α)and the cyclic autocorrelation function R_(x) ^(α), if the cyclicautocorrelation function could be practically generated repeatedtime-to-frequency transforms would be unnecessary. Practical generationof the cyclic autocorrelation function may be accomplished by a photoniccyclic autocorrelation processor and may be a superior solution fornoise-signal separation.

FIG. 2 is a block diagram of a photonic assisted cyclic autocorrelationprocessor 200 in accordance with various aspects of the presentdisclosure. The photonic cyclic autocorrelation processor 200 mayprovide means to solve the Wigner equation (i.e., equation (1)) andvariations thereof in the optical domain without performing therepetitive calculations typically required to obtain a solution.Referring to FIG. 2, the photonic cyclic autocorrelation processor 200may include a master laser 210, a first optical modulator 230, adispersive element 240, a second optical modulator 250, and an opticalcomb filter 260. The master laser 210 may be, for example, but notlimited to, a low-linewidth semiconductor laser or other laser. Themaster laser 210 may generate a self-referenced optical frequency comb220 of signals at different optical wavelengths (i.e., colors, ortones). The tones of the optical frequency comb signal 220 may bemodulated at the first optical modulator 230 by an input signal x(t)225, for example, a radio frequency (RF) signal or other signal,received by a first receiver (not shown). The first optical modulator230 may generate N spectral copies 235 of the input signal x(t) 225 onthe tones of the optical frequency comb signal 220.

The N spectral copies 235 of the input signal x(t) 225 may be sent tothe dispersive element 240. The dispersive element 240 may be an opticalfiber dispersive element, for example, but not limited to, a single modefiber or other dispersive element. The dispersive element 240 maygenerate wavelength-dependent time delays, τ_(k), between adjacentspectral copies of the input signal x(t) 225. The time delays correspondto phase shifts in the frequency domain. The delayed spectral copies(i.e., x(t−τ₁) . . . x(t−τ_(N))) 245 of the input signal x(t) 225 may besent to the second optical modulator 250.

The second optical modulator 250 may modulate the delayed spectralcopies 245 of the input signal x(t) 225 with a conjugate x*(t) 227 ofthe input signal x(t) 225. The second optical modulator 250 may generateconjugate modulated spectral copies (i.e., x*(t)x(t−τ₁) . . .x*(t)x(t−τ_(N))) 255 of the delayed spectral copies 245. The conjugatemodulation of the delayed spectral copies 245 of the input signal x(t)225 results in the practical generation of the terms

${x\left( {t - \frac{\tau}{2}} \right)}{x^{*}\left( {t + \frac{\tau}{2}} \right)}$

from equation (1) that are integrated to obtain the cyclicautocorrelation coefficients.

The conjugate modulated spectral copies 255 may be sent to the opticalcomb filter 260. The optical comb filter 260 may be, for example, butnot limited to, a fine resolution etalon or other optical comb filter.The optical comb filter 260 may perform the integration of the conjugatemodulated spectral copies 255 to produce integrated conjugate modulatedspectral copies 265 corresponding to the cyclic autocorrelationcoefficients for all time delays τ_(k). For example, as illustrated inFIG. 2, the N-th integrated conjugate modulated spectral copy 266 mayhave a complete set of cyclic autocorrelation coefficients R_(x) ^(α)^(i) (τ_(N)) corresponding to τ_(N) delay at each cyclic frequencyα_(i), where i=1, 2, . . . N. Consequently, a full complement of Nintegrated spectral copies will form a complete cyclic autocorrelationtable, eliminating a need for full-rate FFT computations in eitherphotonic or electronic domains.

In some embodiments, the conjugate modulated spectral copies 255 may bemodulated again with the delayed version of the input signal 225 andthen passed through the comb filter 260 to generate computation-freehigher order cumulants (i.e., higher-order cyclic autocorrelationcoefficients). The cyclic autocorrelation function, which is a secondorder process, can discriminate cyclostationary signals contaminatedwith high level noise. There are, however, exceptional signal modulationformats that are immune to second order processes such as spectralcorrelation or cyclic autocorrelation. Signals having these types ofmodulation formats are specially synthesized to be concealed fromcyclostationary receivers. Higher order processes such as third othercumulant analysis can extract those types of signals out of noise.However, the processing complexity increases approximately quadraticallywith the order of the process; therefore, higher order cumulants aredifficult to generate with electronics even for low bandwidth signals.

FIG. 3 is a block diagram of a cyclic autocorrelation coefficientsreadout circuit 300 in accordance with various aspects of the presentdisclosure. Referring to FIG. 3, the readout circuit 300 may include aradio frequency (RF) oscillator 310, a third optical modulator 320, forexample, a single side band (SSB) modulator or other modulator, anoptical wavelength demultiplexer 330, a 90° hybrid optical module 340,and a plurality of detectors D₁-D_(N) 360. The plurality of detectorsD₁-D_(N) 360 may be coherent receivers. The RF oscillator 310 maygenerate a swept frequency. The signal generated by the master laser 210and the swept frequency generated by the RF oscillator 310 may be inputto the third optical modulator 320.

The third optical modulator 320 may shift the signal generated by themaster laser 210 in frequency, for example, by several gigahertz oranother amount, this frequency shifted signal may be used to generatethe second optical frequency comb 326. The second optical frequency comb326 may have the same frequency pitch as the original frequency comb 220used to spectrally clone the input signal 225; however, the comb teethof the second optical frequency comb 326 may be sweeping by severalgigahertz following the swept RF oscillation 310. The generatedfrequency comb may be referred to herein as swept optical samplingsignals 325. The swept optical sampling signals 325 may sweep at a lowerrate than the frequency of the master laser signal, for example, afrequency of approximately 25 kHz or another frequency.

The swept optical sampling signals 325 and the integrated conjugatemodulated spectral copies 265 may be combined by the 90° hybrid opticalmodule 340. The 90° hybrid optical module 340 may act as a coherentreceiver and output four signals: a modulated signal plus localoscillator signal 345 a, a modulated signal minus local oscillatorsignal 345 b, a modulated signal plus conjugate of local oscillatorsignal 345 c, and a modulated signal minus conjugate of local oscillatorsignal 345 d. The output signals of the hybrid optical module 340 may beinput to the optical wavelength demultiplexer 330.

The optical wavelength demultiplexer 330 may include a plurality ofdemultiplexer modules. In some implementations, four demultiplexermodules may be used. Each demultiplexer module may be configured todemultiplex one output 345 a-d of the hybrid optical module 340. Thedemultiplexed signals may be detected by the plurality of detectors 360.The plurality of detectors D₁-D_(N) 360 may be coherent detectors. Insome embodiments, each coherent detector may include two balanceddetectors, with each balanced detector having two PIN diodes (i.e., atotal of four PIN diodes for each coherent detector). Each of theplurality of detectors D₁-D_(N) 360 receives a signal from each of theplurality of demultiplexer modules. For example, for implementationsusing four demultiplexer modules, each detector D₁-D_(N) receives asignal from each demultiplexer module, i.e., each detector D₁-D_(N)receives four signals.

Each of the detectors D₁-D_(N) may coherently detect cyclicautocorrelation coefficients for all time delays τ_(k) of an associatedtone from the integrated conjugate modulated spectral copies 265. Thedetectors D₁-D_(N) may simultaneously stream the coefficients such thatthe R_(x) ^(α) ¹ (τ₁), R_(x) ^(α) ¹ (τ₂), . . . R_(x) ^(α) ¹ (τ_(N))coefficients are detected at the same time, the R_(x) ^(α) ² (τ₁), R_(x)^(α) ² (τ₂), . . . R_(x) ^(α) ² (τ_(N)) are streamed simultaneouslyafter the R_(x) ^(α) ¹ (τ₁), R_(x) ^(α) ¹ (τ₂), . . . R_(x) ^(α) ¹(τ_(N)) coefficients, etc. The detectors D₁-D_(N) may output thedetected cyclic autocorrelation coefficients 335 as a function of time.The detected cyclic autocorrelation coefficients 335 may be digitized,for example using an analog-to digital (A/D) converter, and the spectralcorrelation function calculated.

FIG. 4 is a block diagram of a signal conjugate generator 400 includedin the photonic cyclic autocorrelation processor 200 in accordance withvarious aspects of the present disclosure. Referring to FIG. 4, theconjugate generator 400 may include a fourth modulator 415, and a singlequadrature coherent receiver 440. The single quadrature coherentreceiver 440 may include a directional coupler 430 and a balancedphotodiode 435. The desired RF signal x(t) may be imprinted on areference optical carrier 410 at certain wavelength (i.e., λ₁) using thefourth modulator 415 to generate a modulated signal 420. A secondoptical carrier 425 at a shifted wavelength may be used as a localoscillator to coherently detect the modulated signal 420 at anup-converted frequency and generate the signal conjugate x*(t) 450. Oneof ordinary skill in the art will appreciate that other configurationsfor a conjugate generator may be used without departing from the scopeof the present disclosure.

FIG. 5 is a flowchart of a method 500 for generating cyclicautocorrelation coefficients in accordance with various aspects of thepresent disclosure. Referring to FIG. 5, at block 510 an opticalfrequency comb may be generated. For example, the master laser 210 maygenerate a self-referenced optical frequency comb 220. At block 520, thetones of the optical frequency comb may be modulated with an RF signal.For example, the tones of the optical frequency comb 220 may bemodulated at the first optical modulator 230 by an input signal x(t) 225received by a first receiver (not shown). The first optical modulator230 may generate N spectral copies 235 of the input signal x(t) 225 onthe tones of the optical frequency comb 220.

At block 530, the spectral copies of the input signal may be delayed.For example, the N spectral copies 235 of the input signal may be sentto the dispersive element 240. The dispersive element 240 may generatewavelength-dependent time delays, τ_(k), between adjacent spectralcopies 235 of the input signal. At block 540, the delayed spectralcopies of the input signal may be modulated with the conjugate of the RFsignal. For example, the delayed spectral copies 245 of the input signalx(t) 225 may be sent to the second optical modulator 250. The secondoptical modulator 250 may modulate the delayed spectral copies 245 witha conjugate x*(t) 227 of the input signal x(t) 225. The second opticalmodulator 250 may generate conjugate modulated spectral copies 255 ofthe delayed spectral copies 245.

At block 550, the conjugate modulated delayed spectral copies may beintegrated. For example, conjugate modulated spectral copies 255 may besent to the optical comb filter 260. The optical comb filter 260 mayperform the integration of the conjugate modulated delayed spectralcopies 255 to produce integrated conjugate modulated delayed spectralcopies 265 corresponding to the cyclic autocorrelation coefficientsR_(x) ^(α) ^(i) (τ_(k)) for all time delays τ_(k). A full complement ofN integrated spectral copies will form a complete cyclic autocorrelationtable.

It should be appreciated that the specific steps illustrated in FIG. 5provide a particular method of generating cyclic autocorrelationcoefficients according to an embodiment of the present invention. Othersequences of steps may also be performed according to alternativeembodiments. For example, alternative embodiments of the presentinvention may perform the steps outlined above in a different order.Moreover, the individual steps illustrated in FIG. 5 may includemultiple sub-steps that may be performed in various sequences asappropriate to the individual step. Furthermore, additional steps may beadded or removed depending on the particular applications. One ofordinary skill in the art would recognize many variations,modifications, and alternatives.

FIG. 6 is a flowchart of a method 600 for reading out cyclicautocorrelation coefficients in accordance with various aspects of thepresent disclosure. Referring to FIG. 6, at block 610 swept opticalsampling signals may be generated. For example, the signal generated bythe master laser 210 and the swept frequency generated by the RFoscillator 310 may be input to the third optical modulator 320. Thethird optical modulator 320 may shift the signal generated by the masterlaser 210 to produce a second optical frequency comb 326 and the sweptoptical sampling signals 325.

At block 620, the swept optical sampling signals may be combined withthe integrated conjugate modulated delayed spectral copies of the inputsignal. For example, the swept optical sampling signals 325 and theintegrated conjugate modulated delayed spectral copies 265 may becombined by the 90° hybrid optical module 340 such that each of theswept optical sampling signals 325 samples the corresponding integratedconjugate modulated delayed spectral copy 265. The output of the 90°hybrid optical module 340 may be input to the optical wavelengthdemultiplexer 330. At block 630, the cyclic autocorrelation coefficientsmay be detected. For example, each of the detectors D₁-D_(N) maycoherently detect the cyclic autocorrelation coefficients for all timedelays τ_(k) of an associated tone from the integrated spectral copies265. At block 640, the detectors D₁-D_(N) may output the detected cyclicautocorrelation coefficients 335.

It should be appreciated that the specific steps illustrated in FIG. 6provide a particular method of reading out cyclic autocorrelationcoefficients according to an embodiment of the present invention. Othersequences of steps may also be performed according to alternativeembodiments. For example, alternative embodiments of the presentinvention may perform the steps outlined above in a different order.Moreover, the individual steps illustrated in FIG. 6 may includemultiple sub-steps that may be performed in various sequences asappropriate to the individual step. Furthermore, additional steps may beadded or removed depending on the particular applications. One ofordinary skill in the art would recognize many variations,modifications, and alternatives.

The methods 500 and 600, respectively, may be embodied on anon-transitory computer readable medium, for example, but not limitedto, a memory or other non-transitory computer readable medium known tothose of skill in the art, having stored therein a program includingcomputer executable instructions for making a processor, computer, orother programmable device execute the operations of the methods.

FIG. 7 is a block diagram of a two pumps parametric mixer embodiment 700of a photonic assisted cyclic autocorrelation processor in accordancewith various aspects of the present disclosure. Referring to FIG. 7,comb tones 710 generated by a first laser 715 may be first modulated bythe input signal x(t) 720 by a first optical modulator 725 to generate Nspectral copies of the input signal x(t) 720. The N spectral copies ofthe input signal x(t) 720 may be sent to a dispersive medium 730 togenerate wavelength-dependent delays τ_(k) between adjacent copies 735.

A second optical modulator 745 may modulate a first optical signal froma first blue pump 740 by the conjugate x*(t) of the input signal toserve as the first pump seeds for a two-pumps parametric mixer 760. Asecond optical signal from a second red pump 750 may be sent as acontinuous wave (CW) second pump into the two-pumps parametric mixer760. The two-pumps parametric mixer 760 may be a nonlinear opticalmaterial. In presence of high intensity light power (i.e., the pumps)and phase matching conditions the nonlinear optical material maygenerate a four wave mixing process that leads to idler wavesgeneration. The two-pumps parametric mixer 760 receives the delayedspectral copies 748 as an inner-band input signal and generates anoutput signal 765 that is a combination of the parametrically amplifieddelayed spectral copies 735 and a newly generated idler wavelength bandcreated as a product of the two pumps 742,747, and the parametricallyamplified delayed spectral copies 735 optical fields.

The output of the parametric mixer 760 may be sent to the optical combfilter 762. The optical comb filter 762 may be, for example, but notlimited to, a fine resolution etalon or other optical comb filter. Theoptical comb filter 762 may integrate the idler waves generated at theoutput of the parametric mixer 760. After processing by the parametricmixer 760, only the newly created idler waves, which are the productterms in the equation (1), are important. The other waves (i.e., thepumps and the delayed signal copies) may be filtered out. Since theidler waves are the results of the product of the two pumps and thedelayed signal copies, the one CW pump is constant and has no effect.The other pump carries the conjugate of the input signal that ismultiplied by the delayed signal copies. As a result, the k-th idlerwavelength 769 of the output signal 767 from the comb filter 762 has acomplete set of cyclic autocorrelation coefficients R_(x) ^(α) ^(i)′(τ_(k)), corresponding to τ_(k) delay, where i=1, 2, . . . N.Consequently, a full complement of N idler wavelength copies will form acomplete cyclic autocorrelation table, eliminating a need for full-rateFFT in either photonics or electronic domains.

In accordance with various aspects of the present disclosure, the secondoptical signal from the second red pump 750 may also be modulated togenerate computation-free cumulants (i.e., higher-order cyclicautocorrelation coefficients). As an example, if the second red pump 750is modulated by x(t−t_(k)), the resulting product series will becomposed of x(t−τ_(k))x*(t)x(t−τ₁) terms. Modulation of the secondoptical signal from the second red pump 750 may result in a third orderprocess, rather than a second order process, for the product of thedelayed signal and the conjugate of the input signal.

One of ordinary skill in the art will appreciate that the Wignerfunction has many applications in addition to determining the cyclicautocorrelation function, for example, but not limited to, determining across-ambiguity function (CAF), as well as applications to other aspectsof signal processing, quantum physics, etc. The Wigner function is ageneral technique for time-frequency analysis of signals or equivalentlyposition-momentum analysis. Various applications of the Wigner functionwill benefit from its determination using the reduced computationoptical embodiments of the present disclosure.

In one example, the CAF may be used to determine a location and/orvelocity of a radio frequency transmitter based on signals received froma transmitter by different collectors at different locations. Thelocation of radio frequency transmitters is critical to numerousapplications, for example, but not limited to, military aircraftoperating in hostile areas. FIG. 8 is a diagram illustrating anapplication of the CAF in accordance with various aspects of the presentdisclosure. Referring to FIG. 8, a radio frequency transmitter 810, forexample, but not limited to a radar transmitter, may transmit a signalS(t) to locate one or more targets, for example, one or more aircraft.The transmitted signal may be a target reflection of typical broadcastradiations such as television broadcast signals.

A first receiver 815 (e.g., a first aircraft) disposed at a firstlocation may receive the signal S(t) transmitted by the transmitter 810delayed by a first amount t₁ and having a first frequency ω₁ (i.e.,S(t−t₁)e^(−jω) ¹ ^(t)), where ω₁=2πf₁t. A second receiver 820 (e.g., asecond aircraft) disposed at a second location may receive the signalS(t) transmitted by the transmitter 810 delayed by a second amount t₂and having a second frequency ω₂ (i.e., S(t−t₂)e^(−jω) ² ^(t)), whereω₂=2πf₂t. A data link 825 between the two receivers may enable bothreceivers to know the received signal characteristics. The CAF may beused to jointly compute the Time Difference of Arrival (TDOA),determined as the time difference t₂−t₁ of the received signals, and theFrequency Difference of Arrival (FDOA), determined as the frequencydifference ω₂−ω₁ of the received signals, between two receivers toenable location of the transmitter. Geolocationing techniques such asthis are referred to as passive-radar techniques.

FIG. 9 is a block diagram of a photonic assisted correlation processor900 configured for determining a cross ambiguity function (CAF) inaccordance with various aspects of the present disclosure. Manygeolocation methods utilize the TDOA and FDOA between two receiverscollecting the same transmission. One method of computing the TDOA andFDOA jointly is the cross ambiguity function (CAF). The cross ambiguityfunction is shown in equation (5):

CAF(τ,f)=∫_(−∞) ^(∞) S _(i)(t)S ₂*(t+τ)e ^(−j2πft) dt  (5)

where S₁ and S₂ are signals received from a single transmission sourcethrough two separate receivers, τ is time delay, and f is the frequencydifference. The magnitude of the CAF, or |CAF(τ,f)|, will peak when τand f are equal to the embedded TDOA and FDOA, respectively, between thetwo signals.

Referring to FIG. 9, the photonic assisted correlation processor 900 mayinclude a master laser 910, a first optical modulator 930, a dispersiveelement 940, a second optical modulator 950, and an optical comb filter960. The master laser 910 may be, for example, but not limited to,low-linewidth semiconductor laser or other lasers. The master laser 910may generate a self-referenced optical frequency comb signal 920. Tonesof the optical frequency comb signal 920 may be modulated at the firstoptical modulator 930 by a first signal S₁(t) 925, for example, a radiofrequency (RF) signal or other signal, received by a first receiver (notshown). The first optical modulator 930 may generate N spectral copies935 of the first signal S₁(t) 925 on the tones of the optical frequencycomb 920.

The N spectral copies 935 of the first signal S₁(t) 925 may be sent tothe dispersive element 940. The dispersive element 940 may be an opticalfiber dispersive element, for example, but not limited to, a single modefiber or other dispersive element. The dispersive element 940 maygenerate wavelength-dependent time delays, τ_(k), between adjacentspectral copies of the first signal S₁(t) 925. The time delayscorrespond to phase shifts in the frequency domain. The delayed spectralcopies 945 of the first signal S₁(t) 925 may be sent to the secondoptical modulator 950.

The second optical modulator 950 may modulate the delayed spectralcopies 945 of the first signal S₁(t) 925 with a conjugate S₂*(t) 927 ofa second signal received by a second receiver (not shown) from a sametransmission source as the first signal S₁(t). The conjugate S₂*(t) 927of the second signal may be generated, for example, by the conjugategenerator described with respect to FIG. 4 of the present disclosure orby another method. The second optical modulator 950 may generateconjugate modulated spectral copies 955 of the delayed spectral copies945. The conjugate modulation of the delayed spectral copies 945 of thefirst signal S₁(t) 925 results in the practical generation of the termsS₁(t)S₂*(t+τ) from equation (5) that are integrated to obtain the CAFcorrelation coefficients.

The conjugate modulated delayed spectral copies 955 may be sent to theoptical comb filter 960. The optical comb filter 960 may be, forexample, but not limited to, a fine resolution etalon or other opticalcomb filter. The optical comb filter 960 may perform the integration ofthe conjugate modulated delayed spectral copies 955 to produceintegrated conjugate modulated delayed spectral copies 965 correspondingto the CAF correlation coefficients for all time delays τ_(k). Forexample, as illustrated in FIG. 9, the N-th integrated conjugatemodulated delayed spectral copy 966 may have a complete set of CAFcorrelation coefficients CAF(τ_(N), f_(i)) corresponding to τ_(N) delay,at each frequency f_(i), where i=1, 2, . . . N. A full complement of Nintegrated spectral copies will form a complete CAF correlation table,eliminating a need for full-rate DFT in either photonic or electronicdomains.

A readout circuit, for example, the readout circuit described withrespect to FIG. 3, may optically demultiplex and coherently detect CAFcorrelation coefficients for all time delays τ_(k) of an associated tonefrom the integrated spectral copies. Operation of the readout circuithas been described with respect to FIG. 3 and so will not be repeatedhere. The detectors D₁-D_(N) may output the detected CAF correlationcoefficients as a function of time. The detected CAF correlationcoefficients may be digitized, for example, using an A/D converter.

FIG. 10 is a flowchart of a method 1000 for generating cross-ambiguityfunction correlation coefficients in accordance with various aspects ofthe present disclosure. Referring to FIG. 10, at block 1010 an opticalfrequency comb may be generated. For example, the master laser 910 maygenerate a self-referenced optical frequency comb 920. At block 1020,the tones of the optical frequency comb may be modulated with a first RFsignal. For example, the tones of the optical frequency comb 920 may bemodulated at the first optical modulator 930 by a first signal S₁(t) 925received by a first receiver (not shown). The first optical modulator930 may generate N spectral copies 935 of the first signal S₁(t) 925 onthe tones of the optical frequency comb 920.

At block 1030, the spectral copies of the first signal may be delayed.For example, the N spectral copies 935 of the first signal S₁(t) 925 maybe sent to the dispersive element 940. The dispersive element 940 maygenerate wavelength-dependent time delays, τ_(k), between adjacentspectral copies 935 of the first signal S₁(t) 925. At block 1040, thedelayed spectral copies of the first signal S₁(t) 925 may be modulatedwith a conjugate S₂*(t) 927 of a second signal received by a secondreceiver (not shown) from a same transmission source as the first signalS₁(t). For example, the delayed spectral copies 945 of the first signalS₁(t) 925 may be sent to the second optical modulator 950. The secondoptical modulator 950 may modulate the delayed spectral copies 945 witha conjugate S₂*(t) 927 of a second signal received by a second receiver(not shown) from a same transmission source as the first signal S₁(t).The second optical modulator 950 may generate conjugate modulatedspectral copies 955 of the delayed spectral copies 945.

At block 1050, the conjugate modulated delayed spectral copies may beintegrated. For example, conjugate modulated delayed spectral copies 955may be sent to the optical comb filter 960. The optical comb filter 960may perform the integration of the conjugate modulated delayed spectralcopies 955 to produce integrated conjugate modulated delayed spectralcopies 965 corresponding to the CAF correlation coefficients for alltime delays τ_(k). For example, as illustrated in FIG. 9, the N-thintegrated conjugate modulated delayed spectral copy 966 may have acomplete set of CAF correlation coefficients CAF(τ_(N), f_(i))corresponding to τ_(N) delay, at each frequency f_(i), where i=1, 2, . .. N. Consequently, a full complement of N integrated conjugate modulateddelayed spectral copies will form a complete CAF correlation table.

It should be appreciated that the specific steps illustrated in FIG. 10provide a particular method of generating cyclic autocorrelationcoefficients according to an embodiment of the present invention. Othersequences of steps may also be performed according to alternativeembodiments. For example, alternative embodiments of the presentinvention may perform the steps outlined above in a different order.Moreover, the individual steps illustrated in FIG. 10 may includemultiple sub-steps that may be performed in various sequences asappropriate to the individual step. Furthermore, additional steps may beadded or removed depending on the particular applications. One ofordinary skill in the art would recognize many variations,modifications, and alternatives.

FIG. 11 is a flowchart of a method 1100 for reading out cross-ambiguitycorrelation coefficients in accordance with various aspects of thepresent disclosure. The cross-ambiguity function correlationcoefficients may be read out, for example, using the apparatus describedwith respect to FIG. 3. Referring to FIG. 11, at block 1110 sweptoptical sampling signals may be generated. For example, the signalgenerated by the master laser 910 and the swept frequency generated bythe RF oscillator 310 may be input to the third optical modulator 320.The third optical modulator 320 may shift the signal generated by themaster laser 910 to produce swept optical sampling signals 325.

At block 1120, the swept optical sampling signals may be combined withthe integrated conjugate modulated delayed spectral copies of the inputsignal. For example, the swept optical sampling signals 325 and theintegrated spectral copies 965 may be combined by the 90° hybrid opticalmodule 340 such that each of the swept optical sampling signals 325samples the corresponding integrated modulated delayed spectral copy965. The output of the 90° hybrid optical module 340 may be input to theoptical wavelength demultiplexer 330. At block 1130, the CAF correlationcoefficients may be detected. For example, each of the detectorsD₁-D_(N) may coherently detect the CAF correlation coefficients for alltime delays τ_(k) of an associated tone from the integrated modulateddelayed spectral copies 965. At block 1140, the detectors D₁-D_(N) mayoutput the detected CAF correlation coefficients.

It should be appreciated that the specific steps illustrated in FIG. 11provide a particular method of reading out CAF correlation coefficientsaccording to an embodiment of the present invention. Other sequences ofsteps may also be performed according to alternative embodiments. Forexample, alternative embodiments of the present invention may performthe steps outlined above in a different order. Moreover, the individualsteps illustrated in FIG. 11 may include multiple sub-steps that may beperformed in various sequences as appropriate to the individual step.Furthermore, additional steps may be added or removed depending on theparticular applications. One of ordinary skill in the art wouldrecognize many variations, modifications, and alternatives.

The methods 1000 and 1100, respectively, may be embodied on anon-transitory computer readable medium, for example, but not limitedto, a memory or other non-transitory computer readable medium known tothose of skill in the art, having stored therein a program includingcomputer executable instructions for making a processor, computer, orother programmable device execute the operations of the methods.

FIG. 12 is a block diagram of a two pumps parametric mixer embodiment1200 of a photonic assisted cyclic autocorrelation processor configuredfor determining a cross ambiguity function (CAF) in accordance withvarious aspects of the present disclosure. The two pumps parametricmixer embodiment 1200 of FIG. 12 includes the elements of the two pumpsparametric mixer embodiment 700 illustrated and described with respectto FIG. 7 and are numbered accordingly. The elements of the a two pumpsparametric mixer embodiment 1200 of FIG. 12 operate in the same manneras the elements of the two pumps parametric mixer embodiment 700 of FIG.7 and so will not be further described here.

Referring to FIG. 12, to calculate the CAF, tones of the opticalfrequency comb signal 710 may be modulated at the first opticalmodulator 725 by a first signal S₁(t) 1220, for example, a radiofrequency (RF) signal or other signal, received by a first receiver (notshown). The second optical modulator 745 may modulate the optical signalfrom the first blue pump 740 with a conjugate S₂*(t) of a second signalreceived by a second receiver (not shown) from a same transmissionsource as the first signal S₁(t). The conjugate S₂*(t) 927 of the secondsignal may be generated, for example, by the conjugate generatordescribed with respect to FIG. 4. Processing of the modulated signals bythe embodiment 1200 of FIG. 12 is the same as the processing of themodulated signals explained with respect to the embodiment 700 of FIG. 7and will not be further described here.

FIG. 13 is a flowchart of a method 1300 for generating and reading outWigner function coefficients in accordance with various aspects of thepresent disclosure. The Wigner function coefficients may be generated,for example, using the apparatus described with respect to FIG. 9. TheWigner function coefficients may be read out, for example, using theapparatus described with respect to FIG. 3. Referring to FIG. 13, atblock 1310 an optical frequency comb may be generated. For example, themaster laser 910 may generate a self-referenced optical frequency comb920. At block 1320, the tones of the optical frequency comb may bemodulated with a first signal. For example, the tones of the opticalfrequency comb 920 may be modulated at the first optical modulator 930by a first signal received by a first receiver (not shown). The firstoptical modulator 930 may generate N spectral copies 935 of the firstsignal on the tones of the optical frequency comb 920.

At block 1330, the spectral copies of the first signal may be delayed.For example, the N spectral copies 935 of the first signal may be sentto the dispersive element 940. The dispersive element 940 may generatewavelength-dependent time delays, τ_(k), between adjacent spectralcopies 935 of the first signal. At block 1340, the delayed spectralcopies of the first signal may be modulated with a second signalreceived by a second receiver (not shown). For example, the delayedspectral copies 945 of the first signal may be sent to the secondoptical modulator 950. The second optical modulator 950 may modulate thedelayed spectral copies 945 with a second signal received by a secondreceiver (not shown). The second optical modulator 950 may generateconjugate modulated spectral copies 955 of the delayed spectral copies945.

At block 1350, the modulated delayed spectral copies may be integrated.For example, modulated delayed spectral copies 955 may be sent to theoptical comb filter 960. The optical comb filter 960 may perform theintegration of the modulated delayed spectral copies 955 to produceintegrated modulated delayed spectral copies 965 corresponding to theWigner coefficients for all time delays τ_(k). For example, asillustrated in FIG. 9, the N-th integrated spectral copy 966 may have acomplete set of Wigner coefficients corresponding to τ_(N) delay, ateach frequency f_(i), where i=1, 2, . . . N. Consequently, a fullcomplement of N integrated modulated delayed spectral copies will form acomplete Wigner coefficient table.

At block 1360 swept optical sampling signals may be generated. Forexample, the signal generated by the master laser 910 and the sweptfrequency generated by the RF oscillator 310 may be input to the thirdoptical modulator 320. The third optical modulator 320 may shift thesignal generated by the master laser 910 to produce swept opticalsampling signals 325.

At block 1370, the swept optical sampling signals may be combined withthe integrated modulated delayed spectral copies of the input signal.For example, the swept optical sampling signals 325 and the integratedconjugate modulated delayed spectral copies 265 may be combined by the90° hybrid optical module 340 such that each of the swept opticalsampling signals 325 samples the corresponding integrated modulateddelayed spectral copy 965. The output of the 90° hybrid optical module340 may be input to the optical wavelength demultiplexer 330. At block1380, the Wigner coefficients may be detected. For example, each of thedetectors D₁-D_(N) may coherently detect the Wigner coefficients for alltime delays τ_(k) of an associated tone from the integrated modulateddelayed spectral copies 965. At block 1390, the detectors D₁-D_(N) mayoutput the detected Wigner coefficients.

It should be appreciated that the specific steps illustrated in FIG. 13provide a particular method of generating and reading out Wignercoefficients according to another embodiment of the present invention.Other sequences of steps may also be performed according to alternativeembodiments. For example, alternative embodiments of the presentinvention may perform the steps outlined above in a different order.Moreover, the individual steps illustrated in FIG. 13 may includemultiple sub-steps that may be performed in various sequences asappropriate to the individual step. Furthermore, additional steps may beadded or removed depending on the particular applications. One ofordinary skill in the art would recognize many variations,modifications, and alternatives.

The method 1300 may be embodied on a non-transitory computer readablemedium, for example, but not limited to, a memory or othernon-transitory computer readable medium known to those of skill in theart, having stored therein a program including computer executableinstructions for making a processor, computer, or other programmabledevice execute the operations of the methods.

One of ordinary skill in the art will appreciate that othermodifications to the apparatuses and methods of the present disclosuremay be made for implementing various applications of the Wigner functionwithout departing from the scope of the present disclosure.

The examples and embodiments described herein are for illustrativepurposes only. Various modifications or changes in light thereof will beapparent to persons skilled in the art. These are to be included withinthe spirit and purview of this application, and the scope of theappended claims, which follow.

What is claimed is:
 1. An apparatus for generating a set of cyclicautocorrelation coefficients of an input signal, the apparatuscomprising: a master laser configured to generate an optical frequencycomb signal; a first optical modulator configured to modulate theoptical frequency comb signal with an input signal to generate aplurality of spectral copies of the input signal; a dispersive elementconfigured to delay the plurality of spectral copies of the input signalby a wavelength-dependent time delay; a second optical modulatorconfigured to modulate the delayed plurality of spectral copies with aconjugate of the input signal; and an optical comb filter configured tointegrate the conjugate modulated delayed plurality of spectral copiesof the input signal to generate the set of cyclic autocorrelationcoefficients.
 2. The apparatus of claim 1, wherein the first opticalmodulator and the second optical modulator comprise amplitudemodulators.
 3. The apparatus of claim 1, further comprising: a localoscillator configured to generate a swept frequency; a third opticalmodulator configured to modulate an optical signal generated by themaster laser with the swept frequency from the local oscillator togenerate swept optical sampling signals; a wavelength demultiplexerconfigured to combine the swept optical sampling signals and theintegrated conjugate modulated delayed plurality of spectral copies ofthe input signal such that each of the swept optical sampling signalssamples a corresponding integrated conjugate modulated delayed spectralcopy of the input signal; and a plurality of detectors configured todetect the set of cyclic autocorrelation coefficients for all timedelays of an associated tone from the integrated conjugate modulateddelayed plurality of spectral copies of the input signal, and output thedetected set of cyclic autocorrelation coefficients.
 4. The apparatus ofclaim 3, wherein the third optical modulator is a single side bandmodulator.
 5. The apparatus of claim 3, wherein the plurality ofdetectors are coherent receivers.
 6. A method for generating a set ofcyclic autocorrelation coefficients of an input signal, the methodcomprising: generating an optical frequency comb signal from an opticalsignal generated by a master laser; generating a plurality of spectralcopies of the input signal by modulating the optical frequency combsignal with the input signal; delaying the plurality of spectral copiesby a wavelength-dependent time delay; modulating the delayed pluralityof spectral copies with a conjugate of the input signal; and generatingthe set of cyclic autocorrelation coefficients by integrating theconjugate modulated delayed plurality of spectral copies of the inputsignal.
 7. The method of claim 6, wherein the optical frequency combsignal and the delayed plurality of spectral copies are amplitudemodulated.
 8. The method of claim 6, further comprising: generating aswept frequency; modulating the optical signal generated by the masterlaser with the swept frequency to generate swept optical samplingsignals; combining the swept optical sampling signals and the integratedconjugate modulated delayed plurality of spectral copies of the inputsignal such that each of the swept optical sampling signals samples acorresponding integrated conjugate modulated delayed spectral copy ofthe input signal; detecting the set of cyclic autocorrelationcoefficients for all time delays of an associated tone from theintegrated conjugate modulated delayed spectral copies of the inputsignal; and outputting the detected cyclic autocorrelation coefficients.9. The method of claim 8, wherein the optical signal generated by themaster laser is single side band modulated.
 10. The method of claim 8,wherein detecting the set of cyclic autocorrelation coefficientscomprises coherent detection.
 11. An apparatus for generatingcoefficients for a Wigner function in an optical domain, the apparatuscomprising: a master laser configured to generate an optical frequencycomb signal; a first optical modulator configured to modulate theoptical frequency comb signal with a first signal to generate aplurality of spectral copies of the first signal; a dispersive elementconfigured to delay the plurality of spectral copies of the first signalby a wavelength-dependent time delay; a second optical modulatorconfigured to modulate the delayed plurality of spectral copies with asecond signal; and an optical comb filter configured to integrate themodulated delayed plurality of spectral copies of the first signal togenerate a set of Wigner function coefficients.
 12. The apparatus ofclaim 11, wherein the first optical modulator and the second opticalmodulator comprise amplitude modulators.
 13. The apparatus of claim 11,wherein the second signal is a conjugate of the first signal, and theWigner function is a cyclic autocorrelation function.
 14. The apparatusof claim 11, wherein the second signal is a conjugate of a signaldifferent from the first signal received from a same source as the firstsignal, and the Wigner function is a cross-ambiguity function.
 15. Theapparatus of claim 11, further comprising: a local oscillator configuredto generate a swept frequency; a third optical modulator configured tomodulate an optical signal generated by the master laser with the sweptfrequency from the local oscillator to generate swept optical samplingsignals; a wavelength demultiplexer configured to combine the sweptoptical sampling signals and the integrated modulated delayed pluralityof spectral copies of the first signal such that each of the sweptoptical sampling signals samples a corresponding integrated modulateddelayed spectral copy of the first signal; and a plurality of detectorsconfigured to detect the set of Wigner function coefficients for alltime delays of an associated tone from the integrated modulated delayedspectral copies of the first signal, and output the detected set ofWigner function coefficients.
 16. The apparatus of claim 15, wherein thethird optical modulator is a single side band modulator.
 17. Theapparatus of claim 15, wherein the plurality of detectors are coherentreceivers.